Based on the recently developed fifth-order intrinsic aberration of a soft x-ray and vacuum ultraviolet single-element optical system, this paper proposes a fifth-order aberration (including intrinsic aberration and extrinsic aberration) calculation method for this kind of optical system with multiple elements. First, the fifth-order intrinsic aberration expressions of soft x-ray and vacuum ultraviolet multi-element optical systems are studied; second, the extrinsic aberration calculation method of this kind of optical system is discussed, and corresponding calculation expressions are derived; third, to improve aberration calculation accuracy, the modification of aberration expressions due to the transfer relationship of aperture-ray coordinates with second-order accuracy on the reference exit wavefront between adjacent optical elements is obtained. Finally, the resultant aberration expressions are applied to calculate the aberration of two design examples of soft x-ray and vacuum ultraviolet double-element optical systems with large apertures, and their images are compared with ray-tracing results using Shadow software to validate the aberration expressions. The study shows that the accuracy of the aberration expressions derived in this paper is satisfactory.